# Complex torus

In mathematics, a **complex torus** is a particular kind of complex manifold *M* whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian product of some number *N* circles). Here *N* must be the even number 2*n*, where *n* is the complex dimension of *M*.

All such complex structures can be obtained as follows: take a lattice Λ in **C**^{n} considered as real vector space; then the quotient group

**C**^{n}/Λ

is a compact complex manifold. All complex tori, up to isomorphism, are obtained in this way. For *n* = 1 this is the classical period lattice construction of elliptic curves. For *n* > 1 Bernhard Riemann found necessary and sufficient conditions for a complex torus to be an algebraic variety; those that are varieties can be embedded into complex projective space, and are the abelian varieties.

The actual projective embeddings are complicated (see equations defining abelian varieties) when *n* > 1, and are really coextensive with the theory of theta-functions of several complex variables (with fixed modulus). There is nothing as simple as the cubic curve description for *n* = 1. Computer algebra can handle cases for small *n* reasonably well. By Chow's theorem, no complex torus other than the abelian varieties can 'fit' into projective space.